What are some resources to improve practical expertise in machine learning?
Since you are a software engineer, you already have some experience with one or more programming languages, so this is already a good thing. Moreover, you say that you have a strong mathematical background, so you will probably learn certain machine learning concepts more easily.
When it comes to machine learning, there are open source projects that you can contribute to via Github, and this really boosts your skills and experience.
I also am contributing to an open-source conversation agent via Github.
If you are able to understand and contribute to such projects, then implementing ML/DL algorithms is possible in your target language. I myself have been using Python, but that's just a personal choice, and it's been utilized by many AI engineers and data scientists, globally.

Do you think that attempting kaggle challenges from scratch would be
stimulating? Or too confusing? Do you think delivering a personal
project (i.e. not a byproduct of my current job) in which I employ one
of such methods to solve a potentially real problem would be enough to
be relevant in my cv?

Contributing to open-source projects will link you to other engineers in the same field, and this will eventually open you many paths.
Also, try the book: Artificial Intelligence: A Modern Approach Stuart Russel, Peter Norvig, Prentice Hall, 3rd edition, 2009